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A073003 Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant. 13
5, 9, 6, 3, 4, 7, 3, 6, 2, 3, 2, 3, 1, 9, 4, 0, 7, 4, 3, 4, 1, 0, 7, 8, 4, 9, 9, 3, 6, 9, 2, 7, 9, 3, 7, 6, 0, 7, 4, 1, 7, 7, 8, 6, 0, 1, 5, 2, 5, 4, 8, 7, 8, 1, 5, 7, 3, 4, 8, 4, 9, 1, 0, 4, 8, 2, 3, 2, 7, 2, 1, 9, 1, 1, 4, 8, 7, 4, 4, 1, 7, 4, 7, 0, 4, 3, 0, 4, 9, 7, 0, 9, 3, 6, 1, 2, 7, 6, 0, 3, 4, 4, 2, 3, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) sum_{n>=0} (-y)^n n! = KummerU(1,1,1/y)/y.

Decimal expansion of phi(1) where phi(x)=integral(t=0,infinity, e^-t/(x+t) dt ). - Benoit Cloitre, Apr 11 2003

The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... , m=>-1, is intimately related to Gompertz's constant. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A000110 the Bell numbers and A040027 a sequence that was published by Gould, see for more information A163940. - Johannes W. Meijer, Oct 16 2009

The second part of Lagarias (2013) describes various mathematical developments involving Euler's constant, as well as another constant, the Euler-Gompertz constant. [Jonathan Vos Post, Mar 13, 2013]

REFERENCES

Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 171

Bruce C. Berndt, Ramanujan's notebooks Part I, Springer, p. 144-145.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 424-425.

Wall, H. S., Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948, p. 356

LINKS

Robert Price, Table of n, a(n) for n = 0..10000

A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT]. [From R. J. Mathar, Feb 14 2009]

G. H. Hardy, Divergent Series , Oxford University Press, 1949. p. 29. - Johannes W. Meijer, Oct 16 2009

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.

Simon Plouffe, -exp(1)*Ei(-1)

Ed Sandifer, Divergent Series, How Euler Did It, MAA Online, June 2006. - Johannes W. Meijer, Oct 16 2009

Eric Weisstein's World of Mathematics, Gompertz Constant

Eric Weisstein's World of Mathematics, Exponential Integral

FORMULA

phi(1)=e*(sum(k>=1, (-1)^(k-1)/k/k!) - Gamma)=0.596347362323194... where Gamma is the Euler constant.

G = 0.596347... = 1/(1+1/(1+1/(1+2/(1+2/(1+3/(1+3/(1+4/(1+4/(1+5/(1+5/(1+6/(...- Philippe Deléham, Aug 14 2005

From Peter Bala, Oct 11 2012: (Start)

Stieltjes found the continued fraction representation G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The numerators are in A002793 and the denominators in A002720.

Also, 1 - G has the continued fraction representation 1/(3 - 2/(5 - 6/(7 - ... -n*(n+1)/((2*n+3) - ...)))) with convergents beginning [1/3, 5/13, 29/73, 201/501, ...]. The numerators are in A201203 (unsigned) and the denominators are in A000262.

(End)

G = f(1) with f solution to the o.d.e. x^2*f'(x)+(x+1)*f(x)=1 such that f(0)=1. [Jean-François Alcover, May 28 2013]

EXAMPLE

0.59634736232319407434...

MATHEMATICA

RealDigits[N[-Exp[1]*ExpIntegralEi[-1], 105]][[1]]

PROG

(PARI) eint1(1)*exp(1) \\ Charles R Greathouse IV, Apr 23 2013

CROSSREFS

Equals A001113*A099285. - Johannes W. Meijer, Oct 16 2009

Cf. A001620, A000262, A002720, A002793, A201203.

Sequence in context: A134879 A051158 A117605 * A087498 A238200 A201589

Adjacent sequences:  A073000 A073001 A073002 * A073004 A073005 A073006

KEYWORD

cons,nonn

AUTHOR

Robert G. Wilson v, Aug 03 2002

EXTENSIONS

Additional references from Gerald McGarvey, Oct 10 2005

Link corrected by Johannes W. Meijer, Aug 01 2009

STATUS

approved

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Last modified August 27 15:15 EDT 2014. Contains 246143 sequences.